Mr. Smith's class took a short one question quiz. Dwight's calculation is shown below.
x^2=10
x= square root of 10
When the class switched papers to peer grade, Jonathan noticed that he had a different answer compared to Dwight.
Part A: Revise Dwight's calculation.
Part B: Explain why Dwight's original calculation was incorrect.
Could someone perhaps help me step by step to solve this problem? I'm completely lost.
correct steps:
x^2 = 10
± x = √10
x = ± √10
could you please explain that to me?
Sure! Let's step through the problem and revise Dwight's calculation.
Step 1: Original Equation
The original equation given is x^2 = 10. This means that the value of x squared is equal to 10.
Step 2: Taking the Square Root
Dwight's calculation was to take the square root of 10. However, this is not the correct approach. Taking the square root of both sides of the equation is a valid step, but it must be done carefully.
To revise Dwight's calculation, we need to consider the fact that the square root of a number has both a positive and a negative value. So, taking the square root of both sides of the equation, we get two possible solutions: x = +√10 and x = -√10.
Part A: Revised Calculation for Dwight
To revise Dwight's calculation, we need to include both possible solutions:
x = +√10 or x = -√10
This means that the value of x can be either the positive square root of 10 or the negative square root of 10.
Part B: Explanation of Dwight's Mistake
Dwight's original calculation was incorrect because he assumed that there is only one solution to the equation x^2 = 10. However, the fundamental principle of taking the square root is that it yields both positive and negative solutions.
By only considering the positive square root of 10, Dwight overlooked the negative square root, leading to a different answer compared to Jonathan. To obtain the correct answer, it is essential to consider both possible solutions.
By the definition of the √ operator, we get the positive square root of the number
so when we have an equation such as x^2 = 4
there are two answers +2 or -1, BUT the negative answer comes from √(x^2) not from √4
this is best illustrated using an example such as
x^2 < 25, we intuitively know that correct answer is the set of all numbers between -5 and 5
that is, -5 < x < +5
The common error to solve this is this:
x^2 < 25
x < ± 5
which means x < -5 and x < +5, which would include such number as -10
but -10 is NOT a solution to the given inequality
the correct method is to use the ± with the x^2,
x^2 < 25
±x < 5
which means x < 5 and -x < 5
giving us : x < 5 and x > -5 or the correct solution of -5 < x < 5